We propose a new heuristic algorithm for the Maximum Happy Vertices problem, using tree decompositions. Traditionally, such algorithms construct an optimal solution of the given problem instance through a dynamic programming approach. We modify this procedure by integrating a parameter $W$ that dictates the number of dynamic programming states to consider. We drop the exactness guarantee in favour of a shorter running time. However, if $W$ is large enough such that all valid states are considered, our heuristic algorithm proves optimality of the constructed solution. Our algorithm more efficiently constructs an optimal solution for the Maximum Happy Vertices problem than the exact algorithm for graphs of bounded treewidth. Furthermore, our algorithm constructs higher quality solutions than state-of-the-art heuristic algorithms Greedy-MHV and Growth-MHV for instances of which at least 40% of the vertices are initially coloured, at the cost of a larger running time.

翻译：我们用树分解法为最大快乐口号问题提出一种新的超自然算法。 传统上, 这种算法通过动态编程法为给定问题实例构建了最佳解决办法。 我们通过整合参数$W$来修改这个程序, 从而决定需要考虑多少动态编程状态。 我们放弃了精确性保证, 以缩短运行时间为主。 但是, 如果美元足够大,能够考虑所有有效国家, 我们的超自然算法就能证明构建的解决方案是最佳的。 我们的算法能更有效地构建一个最佳解决办法, 解决最大快乐口号问题, 而不是边界树枝图的精确算法。 此外, 我们的算法可以构建比高质量的解决方案, 而不是最先进的超自然算法的Greedy- MHV 和Crecult-MHV 更高质量的解决方案, 用于至少40%的脊椎最初是彩色的, 代价是更大的运行时间。